# Geometric Calculus, Clifford Algebra, and Calculus of Variations

It has always bothered me that I was told in my Calculus of Variations class that it's only possible to minimize a function with respect to one objective. Obviously sometimes it is possible to minimize with respect to more than one objective, depending on the function and the constraints.

I just started reading about Geometric Calculus yesterday, and woke up this morning thinking that geometric calculus should provide a good approach to dealing with those solvable multiobjective optimization problems, if only because of the peoperties of "antimultivectors". In particular, it seems that Geometric Calculus might be just right for describing reduced-dimension manifolds in a space of many variables.

Is there a good paper, or set of papers, that describe such an approach to optimization?

• What is an antimultivector? (I know what multivectors are.) – mr_e_man May 20 '18 at 7:01
• If I understand correctly, an antimultivector is a blade of grade (n-m) where n is the dimensionality of the space. It transforms like a pseudovector. The term "antimultivector" doesn't seem to be used much. – S. McGrew May 20 '18 at 15:19
• Well then, what is $m$? I guess it's the dimension of a submanifold. It seems that an antimultivector is simply a multivector with a certain relation to another one (perhaps by multiplication by the tangent blade). I do see the words with "pseudo" more often than "anti", but I like neither... – mr_e_man May 20 '18 at 17:16
• the definition of 'antivector' is helpful -- wikipedia. – S. McGrew May 20 '18 at 17:35
• en.wikipedia.org/wiki/Blade_(geometry) "In a space of dimension n, a blade of grade n − 1 is called a pseudovector[2] or an antivector." So m would be a different value instead of 1. – S. McGrew May 20 '18 at 17:45